By Noah Barkin

BRUSSELS (Reuters) - Hoping to stave off a broader political crisis after Britain's shock decision to leave the EU, European leaders agreed on Wednesday to spend the next nine months developing proposals for an overhaul of the bloc amid deep divisions between its members.

Disillusion with the EU has risen sharply following years of economic weakness and after a record influx of refugees and series of deadly attacks by Islamic militants.

The problems have fueled the sense that elites in Brussels and other European capitals are ineffective and out of touch with the concerns of ordinary people.

Last week, the anger bubbled over in Britain's Brexit vote, which threw six decades of closer European integration into reverse and raised fears of a domino effect on the continent, where anti-EU, xenophobic parties are on the rise.

EU leaders, who met on Wednesday without Britain, agree that change is needed. But they also know that time is required to get the remaining 27 members behind a common European initiative due to a deep divide over what lessons to draw from Brexit.

German Chancellor Angela Merkel, speaking to reporters at the end of the summit, said it was unrealistic to consider radical changes, such as moving towards a fiscal or political union, in the current environment. These would require changes to the EU's Lisbon Treaty and more referendums, which leaders are desperate to avoid.

"It is not about more or less Europe but about delivering better results," Merkel said.

"Our citizens often don't understand why we are doing something and what our goals are. All of us want to change this. It is not about changing the EU Treaty, about introducing more laws or less. It's about delivering on our goals."

Officials said that Dutch Prime Minister Mark Rutte, who holds the rotating presidency of the EU, had made clear to other leaders that it was not the time for "revolutions".

Another top official acknowledged that vague pledges to create a "better Europe" were largely empty but that the main priority for now was to send a simple message that everyone around the table could agree on.

EXISTENTIAL

The period of "political reflection" will start in earnest in mid-September at a summit in Bratislava, Slovakia. Some EU leaders have said the goal is to reach a set of proposals by March of next year, the 60th anniversary of the EU's founding Rome Treaty.

The period mirrors the one that followed French and Dutch rejections of a European constitution in dual referendums in 2005. Merkel came to power the same year and led negotiations on the more modest Lisbon Treaty.

But the current crisis is more existential for the EU because of the Brexit vote, which in one fell swoop deprives the bloc of one of its only economic and political heavyweights.

Years of crisis have also left deep scars among member states and there is very little agreement about what the changes should entail.

Poland, Slovakia, Hungary and the Czech Republic have called for the powers of the European Commission to be reined in following Brexit.

Politicians in France and Belgium have suggested that a core of like-minded member states press ahead with deeper integration in a "multi-speed" Europe.

And German Finance Minister Wolfgang Schaeuble appears to favor stricter budget rules, an idea that would infuriate southern European countries which are struggling to cope with high unemployment after years of austerity.

A poll by the Pew Research Center earlier this month showed support for the EU plunging in its biggest member states.

The fall was most pronounced in France, where only 38 percent of respondents said they had a favorable view of the EU, down 17 points from last year. Favorability ratings were also down sharply in Spain, at 47 percent, and Germany, at 50 percent.

European Council President Donald Tusk, speaking at the end of the summit, said the flood of over a million migrants into Europe over the past year had been a major factor in the rising euroskepticism.

"Irregular migration was and is one of the principle reasons of this crisis of confidence in Europe," said Tusk, noting that Europe needed to show its citizens that it could control its external borders.

The influx of migrants has fueled populist, anti-immigrant parties like the National Front in France and Freedom Party in Austria, which have cheered the Brexit vote and called for referenda across Europe.

"The Brexit vote risks preventing the EU from developing further," Commerzbank Chief Economist Joerg Kraemer told Reuters. "It has fueled euroskeptic forces in other states and in this situation, the politicians won't dare push for more Europe."

(Additional reporting by Alastair Macdonald & Paul Taylor, EU summit team; Writing by Noah Barkin; Editing by Alastair Macdonald)